PPU 960 Physics Note [Sem 1 : Chapter 1 - Physical Quantities and Units]

00

STPM

2012

About Author:

Facebook: www.facebook.com/groups/josh.lrt Email: [email protected] [Mr. Josh]

Contact No: +6018-397 6808 [Mr. Josh]

Chapter 1 – Physical Quantities and Units By : Josh, LRT

2012 © LRT Documents Copyrighted. All rights reserved. Page 1 of 17

Chapter 1 – Physical Quantities and Units

1.1 Base quantities and SI units

The SI is founded on seven SI base units for seven base quantities assumed to be mutually

independent, as given in the table below:

No Base Quantity SI Unit

Name Symbol Name Symbol

1. Length meter m

2. Mass m kilogram kg

3. Time t second s

4. Electric Current I ampere A

5. Thermodynamic Temperature T kelvin K

6. Amount Of Substance n mole mol

7. Luminous/Light Intensity L candela cd

1.2 Dimensions of physical quantities

Dimensions

To express a quantity in terms of units may not be good enough. For example, unit of speed is ms-1

but for certain countries, the units are miles per hour or fth-1

. To avoid this discrepancy, we express

the quantity in term of its base quantities.

Base Quantity

No Name Symbol for dimension Description

1. length L The one dimensional extent of an object.

2. mass M The amount of matter in an object.

3. time T The duration of an event.

4. electric current I Rate of flow of electrical charge.

5. thermodynamic

temperature

Average energy per degree of freedom of a

system.

6. amount of substance N Number of particles compared to the

number of atoms in 0.012 kg of 12

C.

7. luminous intensity J Amount of energy emitted by a light

source in a particular direction.

Examples:

Find the dimension of “Speed” / “Velocity”.

, - 0

1



Dimensional Analysis

Dimensional analysis is a method of using the known units in a problem to help deduce the process

of arriving at a solution. Besides, it can be also used in checking the correctness of an equation

which you have derived after some algebraic manipulation. These tips will help you apply

dimensional analysis to a problem.

* This required using the Base Quantity to start an analysis from derived quantities.

Examples:

1. Find the dimension of acceleration and its SI Unit.

, - 0

1

2. Find the dimension of pressure and its SI Unit.

, - 0

1

0

1

0

1

[

]

[ ]



Dimensional Homogeneity

It is the quality of an equation having quantities of same units on both sides. A valid equation in

physics must be homogeneous, since equality cannot apply between quantities of different nature.

This can be used to spot errors in formula or calculations.

Examples:

Analyze the equation below and determine whether it is “dimensionally consistence” or

“dimensional inconsistence”.

To do that, remember the substitution:

1. Change Unit Symbol to Dimensional Symbol.

2. Change a constant to 1.

QUESTION 1,

Equation:

, - , -

, - ,

-

, -

, - ( )( )

0

1 ( )( )( )

, - , - 0

1

It is a dimensionally consistence.

QUESTION 2,

Equation: , - , - , - , -

, -

, -

, - ( )( )( )

*, - , -+ , -

It is a dimensionally inconsistence.



Construction empirical equations

It is ability or the strongest part for a PHYSICS student which you or student is able to construct

and develop a new equation based on the experiment in physics.

1. The only way to do it is through a method call “Dimensional Analysis”.

2. Equations must be balanced both side which you can check using “Dimensional Homogeneity”

method.

3. Rename the dimensional equation into a standard unit equation which represent using the

symbol that exist or new proven by you or others.

4. Finally, a new equation is born!

Example:

We want to know how the speed of waves, v; on a string depends its mass, m; length, l; and tension,

Q? We can solve this problem using dimensional analysis method.1

( )( )( )( )

Form T,

− −

Form M,

0

0

−

Form L,

b

b

b

Substitute the value of a, b and c into

the equation we made:

𝑣 𝑘𝑚𝑎𝑙𝑏𝑄𝑐

𝑣 𝑘𝑚 1

𝑙1

𝑄1

𝑣 𝑘 𝑙𝑄

𝑚

𝑣 𝑘 𝑙𝑄

𝑚



1.3 Scalars and vectors

What are scalar quantities and vectors quantities?

Scalars → are quantities that are fully described by a magnitude (or numerical value) alone.

Vectors → are quantities that are fully described by both a magnitude and a direction.

In this chapter, we are going to focus only on vectors and the direction that changing during a

calculation in an equation.

Vectors

A geometric way of representing quantities that have direction as well as magnitude. When vectors

are written, they are represented by a single letter in bold type or with an arrow above the letter,

such as ⃗⃗ .

Vectors can be equal if:

a. Both lines are parallel and the directions are same.

b. Labels are same for both lines.

c. Magnitude of A = Magnitude of B

d. Direction of A = Direction of B

1. Sum of vectors

Sum of vectors is by taking the value on the ⃗⃗⃗ and ⃗⃗ ⃗ and calculate the resultant vector

[ ⃗⃗⃗ + ⃗⃗ ⃗ ] which here are 2 methods to do it:

a. Parallelograms of vectors

[ ⃗ + ⃗⃗⃗⃗ ]

b. Triangle of vectors

[ + ⃗ ]

𝐴

�⃗�

𝐴

�⃗�

𝐶

�⃗�



2. Polygon of vectors

To find the sum of three or more vectors, a polygon is drawn. The parallelogram rule of addition

is partial case of general Polygon Rule used for adding several vectors [ ⃗⃗⃗ + ⃗⃗ ⃗ ].

[ ⃗ ⃗⃗⃗ ⃗⃗ ⃗ ]

3. Subtraction of vectors

Subtracting of vectors ⃗⃗⃗ * + and ⃗⃗ ⃗ * + is vector difference ⃗⃗ ⃗⃗⃗ − ⃗⃗ ⃗ The vector difference is determined by Triangle Method of subtraction.

4. Vector Product of Vectors

Vector (or cross) product of vectors ⃗⃗⃗ * + and ⃗⃗ ⃗ * + is vector ⃗⃗ ⃗⃗⃗ ⃗⃗ ⃗

The vector is normal to plane in which the vectors ⃗⃗⃗ and ⃗⃗ ⃗ lie (plane x-y) directed in

accordance with → Right-Hand Rule.

When right palm is half-bent from ⃗⃗⃗ to ⃗⃗ ⃗ its thumb shows the direction of vector .

Example:

�⃗�

𝑎 𝑏 𝜃

As we stated before, when we find a Vector Product the

result is a vector. We define the modulus, or magnitude, of

this vector as:

�⃗⃗� 𝑟 ⃗⃗⃗ − 𝑟 ⃗⃗ ⃗



So, at this stage, a very similar definition t o the scalar product, except now the sine of θ appears in

the formula. However, this quantity is not a vector. To obtain a vector we need to specify a

direction.

Application of vectors product with its direction

Diagram A → Situation of a screw turn is such a direction from a to b.

The direction is anticlockwise.

Equation used: ̂

It will gains a +ve value for ̂.

Diagram B → Situation of a screw turn is such a direction from b to a.

The direction is clockwise.

Equation used: (− ̂)

It will gains a -ve value for ̂.

Conclusion:

The direction of is opposite to that of .

The vector product is not commutative.

An equation for the opposite direction vector can be written as: − .

By define the direction of the vector product is such that it

is at right angles to both a and b. Figure shows that we

have two choices for such a direction.



Resolving a vector

A vector is only can be solved using the TRIANGLE THEOREM. Which is the basic unit of an

equilibrium acts on different angle 0 .

𝜃

𝐹𝑦



EXAMPLE QUESTION:

Given an object is moving with a constant acceleration. While moving, all the supply forces is not

proportional to the axis, but this made the object to move in a straight direction by having a 3

dimensional orientation forces which is F1, F2 and F3.

Find the angle of resultant force ( º) and its force (N).

Steps of solving:

1. Draw the Cartesian Axis on the diagrams.

2. Find the unknown angle.

3. Try to solving forces acts of Fy = GREEN and Fx = RED

4. Solving by defining Fy = Y-component and Fx = X-component

X-component

*( 00 ) ( 000 )+ − ,( 00 0 )-

*( 0 ) ( 0)+ − , -

Y-component

,−( 00 )- ( 000 ) , ( 00 0 )-

,−( 0)- ( ) ( )



After we solved the x and y component, we will get the forces like below:

5. Let’s get solve the R and the .

TO solve R,

( ) ( )

0

TO solve ,

.

/

Finally, draw and label the resultant force [R].

Resultant Force = R

𝜃

𝑅 0



1.4 Uncertainties in measurements

What are uncertainties and measurements?

Uncertainties → are predictions of physical measurements that already made, or to the unknown.

Measurements → are the process or the result of determining the ratio of a physical quantity.

Type of errors

1. Random Errors

Random errors in experimental measurements are caused by unknown and unpredictable

changes in the experiment. These changes may occur in the measuring instruments or in the

environmental conditions.

Examples of causes of random errors are:

o Electronic noise in the circuit of an electrical instrument.

o Irregular changes in the heat loss rate from a solar collector due to changes in the wind.

The precision of a measurement is how close a number of measurements of the same quantity

agree with each other. The precision is limited by the random errors. It may usually be

determined by repeating the measurements.

2. Systematic Errors

Systematic errors in experimental observations usually come from the measuring instruments.

They may occur because:

There is something wrong with the instrument or its data handling system, or

Because the instrument is wrongly used by the experimenter.

Two types of systematic error can occur with instruments having a linear response:

1. Offset or zero setting error in which the instrument does not read zero when the quantity to

be measured is zero.

2. Multiplier or scale factor error in which the instrument consistently reads changes in the

quantity to be measured greater or less than the actual changes.

Examples of systematic errors caused by the wrong use of instruments are:

Errors in measurements of temperature due to poor thermal contact between the

thermometer and the substance whose temperature is to be found.

Errors in measurements of solar radiation because trees or buildings shade the radiometer.

The accuracy of a measurement is how close the measurement is to the true value of the

quantity being measured. The accuracy of measurements is often reduced by systematic errors,

which are difficult to detect even for experienced research workers.



Common sources of error in physics laboratory experiments:

Incomplete definition (may be systematic or random)

One reason that it is impossible to make exact measurements is that the measurement is not always

clearly defined. For example, if two different people measure the length of the same rope, they

would probably get different results because each person may stretch the rope with a different

tension. The best way to minimize definition errors is to carefully consider and specify the

conditions that could affect the measurement.

Failure to account for a factor (usually systematic)

The most challenging part of designing an experiment is trying to control or account for all possible

factors except the one independent variable that is being analyzed. For instance, you may

inadvertently ignore air resistance when measuring free-fall acceleration or you may fail to account

for the effect of the Earth’s magnetic field when measuring the field of a small magnet. The best

way to account for these sources of error is to brainstorm with your peers about all the factors that

could possibly affect your result. This brainstorm should be done before beginning the experiment

so that arrangements can be made to account for the confounding factors before taking data.

Sometimes a correction can be applied to a result after taking data to account for an error that was

not detected.

Environmental factors (systematic or random)

Be aware of errors introduced by your immediate working environment. You may need to take

account for or protect your experiment from vibrations, drafts, changes in temperature, electronic

noise or other effects from nearby apparatus.

Personal errors (random)

come from carelessness, poor technique, or bias on the part of the experimenter. The experimenter

may measure incorrectly, or may use poor technique in taking a measurement, or may introduce a

bias into measurements by expecting (and inadvertently forcing) the results to agree with the

expected outcome.

Instrument resolution (random)

All instruments have finite precision that limits the ability to resolve small measurement differences.

For instance, a meter stick cannot distinguish distances to a precision much better than about half of

its smallest scale division (0.5 mm in this case). One of the best ways to obtain more precise

measurements is to use a null difference method instead of measuring a quantity directly. Null or

balance methods involve using instrumentation to measure the difference between two similar

quantities, one of which is known very accurately and is adjustable. The adjustable reference

quantity is varied until the difference is reduced to zero. The two quantities are then balanced and

the magnitude of the unknown quantity can be found by comparison with the reference sample.

With this method, problems of source instability are eliminated, and the measuring instrument can

be very sensitive and does not even need a scale.



Physical variations (random)

It is always wise to obtain multiple measurements over the entire range being investigated. Doing so

often reveals variations that might otherwise go undetected. These variations may call for closer

examination, or they may be combined to find an average value.

Parallax (systematic or random)

This error can occur whenever there is some distance between the measuring scale and the

indicator used to obtain a measurement. If the observer's eye is not squarely aligned with the pointer

and scale, the reading may be too high or low (some analog meters have mirrors to help with this

alignment).

Instrument drift (systematic)

Most electronic instruments have readings that drift over time. The amount of drift is generally not

a concern, but occasionally this source of error can be significant and should be considered.

Lag time and hysteresis (systematic)

Some measuring devices require time to reach equilibrium, and taking a measurement before the

instrument is stable will result in a measurement that is generally too low. The most common

example is taking temperature readings with a thermometer that has not reached thermal

equilibrium with its environment. A similar effect is hysteresis where the instrument readings lag

behind and appear to have a "memory" effect as data are taken sequentially moving up or down

through a range of values. Hysteresis is most commonly associated with materials that become

magnetized when a changing magnetic field is applied.

Failure to calibrate or check zero of instrument (systematic)

Whenever possible, the calibration of an instrument should be checked before taking data. If a

calibration standard is not available, the accuracy of the instrument should be checked by

comparing with another instrument that is at least as precise, or by consulting the technical data

provided by the manufacturer. When making a measurement with a micrometer, electronic balance,

or an electrical meter, always check the zero reading first. Re-zero the instrument if possible, or

measure the displacement of the zero reading from the true zero and correct any measurements

accordingly. It is a good idea to check the zero reading throughout the experiment.



A simple idea why we need to provide uncertainties in our measurements?

ANSWER: A mathematician might give the value of , as 3.1415927. A mathematician is

interested in numbers as ideas. But a physicist is interested in measuring the world in order to

explore relationships between one part of it and another.

Making measurement is a practical activity and there is no measuring instrument that can

provide a perfectly precise answer. Practical measurements have uncertainties, which we also

call 'errors'.

It is essential that we take account of these practical uncertainties when we reach conclusions

from practical investigations.

To a physicist, therefore, a measurement of might be written as:

This tells on that we have confidence that the measurement lies between 3.12 and 3.16, but we

cannot be confident of greater precision. Therefore, 0.02 is our uncertainty or error.

SIMPLE question:

How thick is the pile of coins?

𝜋 𝐶

𝐷

( 0 0 0 )

We only know the result to the nearest 0.1cm.

We cannot claim that we know the result to the nearest

0.01cm.

We can write the answer with two digits.

Only two figures are significant.

We know that the result is equal to or greater than 1.35

cm and is less than 1.45 cm.

We can write the result like INS:

0.05 is the uncertainty.



To calculate percentage error:

00

So, what is the percentage error of your reading of ( 0 0 0 ) ?

0 0

0 00

Why is a third digit in these length measurements “insignificant”?

ANSWER: The third digit counts hundredths of centimeters and we cannot know the thickness of

pile of coins with such precision.

Conclusion: There is uncertainty, or possible error, in every measurement. We should record

measurements using an appropriate number of significant figures.

To calculate fractional error:

00

So, what is the fractional error of your reading of ( 0 0 0 ) ?

0 0

0

0 0

Conclusion: A measurement is more accurate if the fractional error is smaller.



Principles for calculating the uncertainty in a derived quantity

1. This is an absolute uncertainty.

2. Mainly used to calculate the units of uncertainty exist in a measurement.

a. Example if 2 readings are: ( ) and ( )

o

o −

The uncertainty will be calculated as .

b. Example if 2 readings are: ( ) and ( )

The uncertainty will be calculated as

.

Plenary:

There is uncertainty in every measurement.

We should record measurements using an appropriate number of significant figures. This

depends on the precision of the measuring process.

We can give values with recognition of our uncertainty. The uncertainty is a possible error

in the value, and is usually called simply the error.

There is random error in every measurement, since no measurement has perfect precision.

Random error can be reduced but not eliminated by good experimental technique.

The effect of random error can be reduced by taking several measurements and taking the

mean, and by other averaging methods including graph-plotting.

Systematic error results in measurements that deviate in a consistent way from the unknown

'true' values.

It is possible to eliminate systematic error by good experimental technique.

We can show the sizes of errors on plotted points on a graph using error bars.

Since error bars show the likely limits of our uncertainty in any measurement, the line of the graph

should lie within the ranges defined by the error bars of all points. Decisions on what is and what is

not a valid line to draw must be based on this.



EXAMPLE QUESTION:

The following measurements were made to determine the density of a metal cylinder.

Diameter of cylinder, ( 0 0 )

Length of cylinder, ( 0 )

Mass of cylinder, ( 0 0)

a) What is the percentage error of the density?

b) Calculate the density of the metal cylinder to correct number of significant figure.

( ( )

)( )

0

( 0 0 )

Percentage error:

𝜌

𝜌 .

Δ𝑚

𝑚 Δ𝑑

𝑑

Δ

/ 00

.

( )

/ 00

Uncertainty:

𝜌

𝜌

0

0